Nested Subgraphs of Complex Networks

We analytically explore the scaling properties of a general class of nested subgraphs in complex networks, which includes the $K$-core and the $K$-scaffold, among others. We name such class of subgraphs $K$-nested subgraphs due to the fact that they generate families of subgraphs such that $\mathrm{.}\mathrm{.}\mathrm{.}{S}_{K+1}(\mathcal{G})\subseteq S_K(\mathcal{G})\subseteq S_{K-1}(\mathcal{G})\mathrm{.}\mathrm{.}\mathrm{.}$. Using the so-called {\em configuration model} it is shown that any family of nested subgraphs over a network with diverging second moment and finite first moment has infinite elements (i.e. lacking a percolation threshold). Moreover, for a scale-free network with the above properties, we show that any nested family of subgraphs is self-similar by looking at the degree distribution. Both numerical simulations and real data are analyzed and display good agreement with our theoretical predictions.